2022/06/19 by Sébastien Boucksom, Boucksom, Sebastien, Mattias Jönsson +1 · 7 citations
Economics, Econometrics and Finance · Mathematics · #32P05 #32Q26 #Algebraic Geometry (math.AG) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Stochastic processes and financial applications #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2206.09492
openalex publication_date 2022/06/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
To any projective pair (X,B) equipped with an ample ℚ-line bundle L (or even any ample numerical class), we attach a new invariant β(μ)∈ℝ, defined on convex combinations μ of divisorial valuations on X, viewed as point masses on the Berkovich analytification of X. The construction is based on non-Archimedean pluripotential theory, and extends the Dervan-Legendre invariant for a single valuation--itself specializing to Li and Fujita's valuative invariant in the Fano case, which detects K-stability. Using our β-invariant, we define divisorial (semi)stability, and show that divisorial semistability implies (X,B) is sublc (i.e. its log discrepancy function is non-negative), and that divisorial stability is an open condition with respect to the polarization L. We also show that divisorial stability implies uniform K-stability in the usual sense of (ample) test configurations, and that it is equivalent to uniform K-stability with respect to all norms/filtrations on the section ring of (X,L), as considered by Chi Li.